What's the difference between 'affirming the consequent' and 'commutation of conditionals'?

Deductively, both forms are always invalid.  From "P then Q and Q", "P" doesn't follow.  Similarly, from "P then Q", "Q then P" doesn't follow.   From "All Swans are White", even a billion swans that are all white doesn't prove that all are.

On the other hand, Inductive reasoning is an attempt to determine what is probably true.   So looking for swans and seeing the first billion are white makes the statement "Most Swans Are White" true.   If you had the argument: "if he was the shooter, his fingerprints would be on the gun.  His fingerprints were on the gun.  So  probably , he was the shooter."  That would be sound inductive reasoning, even though it's essentially affirming the consequent.  But the conclusion is simply-"probably."  It is not absolute proof.  How probable depends on the circumstances (and in some scenarios such reasoning may not even make the conclusion probable).  

commutation of conditionals

Conditionals are not equivalent to their converse. 

Example: If you know logic, then you're good at reading. Therefore, if you're good at reading, then you know logic.

affirming the consequent

The example above only states conditionals.  Neither conditional say something about a person actually knowing logic or being good at reading. They could be viewed as rules. If you know this, then you are good at this. If you're good at this, then you know this. 

For affirming the consequent, we would be actually affirming that someone did something. 

Example: If you know logic, then you're good at reading. Tom is good at reading. Then he must know logic. 

We could write two rules of inference using this context:

(1) Given a rule in the form of a conditional, you cannot infer the converse of that rule. 

(2) Given a conditional and the consequent of that conditional, we cannot infer the antecedent.

In calculus

Commutation of conditionals.

The theorem, if a function is differentiable, then it is continuous, might get confused with the wrong rule, if a function is continuous, then it is differentiable. It's important to stay put for commutation of conditionals in theorems with that form.

Affirming the consequent.

f(x) = |x| is continuous. Therefore, since differentiable functions are continuous, f is differentiable.

In programming

Commutation of conditionals.

Please take the following examples with a grain of salt. I'm doing self-taught programming.

If a variable is allocated in an automatic storage class, then the variable is local. Therefore, if a variable is local, it is allocated in the automatic storage class. I just learned about this, but I think that local variables can also be allocated in a static storage class.

Affirming the consequent.

The developer qualified an identifier as static. But the variable associated with the identifier is local. Therefore, its memory location must be in the automatic storage class.

Hi, Trevor!

A fallacy is a mistake in the course of an argument. In light of that, the difference between the fallacy of affirming the consequent and the fallacy of the commutation of conditionals is that they are mistakes made in the course of two different kinds of arguments. Both arguments start with a premise using a conditional statement, but let’s see how they differ from there.

A premise of the argument committing the fallacy of affirming the consequent states that the consequent of the conditional is true. Notice it has a premise that says Q. Whereas, the argument committing the fallacy of the commutation of conditionals does not have a premise stating that the consequent of the conditional is true. Notice that nowhere does it have a premise that says Q. 

The argument committing the fallacy of affirming the consequent concludes from its premises that the antecedent of the conditional is true. Notice that P is its conclusion. Whereas, the argument committing the fallacy of the commutation of conditionals does not conclude that the antecedent of the conditional is true. Notice that nowhere does the conclusion say that P is true. 

In a nutshell, the argument committing the fallacy of affirming the consequent does not state that Q is sufficient for P, whereas the argument committing the fallacy of the commutation of conditionals does. The argument committing the fallacy of the commutation of conditionals does not state that the consequent is true nor that the antecedent is true, whereas the argument committing the fallacy of the affirming the consequent does. 

Are you able to provide an example of when the affirming the consequent fallacy has been committed but not the commutation of conditionals fallacy? And/Or 

An example of when the commutation of conditionals fallacy has been comitted but not the affirming the consequent fallacy.

Here is an argument that affirms the consequent but does not commutate conditionals: 

1. If it is garbage day (P), then my neighbor has his garbage bin on the end of the driveway (Q)

2. My neighbor has his garbage bin on the end of the driveway (Q)

3. Therefore, it is garbage day (P)

The consequent is asserted and the antecedent is concluded. So, this argument commits the fallacy of affirming the consequent. However, the argument does not conclude that my neighbor has his garbage bin on the end of the driveway is sufficient for it is garbage day. So, the argument does not commit the fallacy of the commutation of conditionals. 

Here is an argument that commutates conditionals but does not affirm the consequent: 

If it is my birthday (P), then my family will bake a cake tonight (Q)

Therefore, if my family will bake a cake tonight (Q), then it is my birthday (P)

The argument does not state that my family will bake a cake nor does it conclude that it is my birthday. Therefore, it does not affirm the consequent. But it does commutate the conditional. 

Thank you, Trevor. 

From, Kaiden